teacher demand model for basic education - ulisboa · 2013 the european commission demanded the...

Teacher Demand Model for Basic Education

Miguel Tavares Aleluia

Thesis to obtain the Master of Science Degree in

Engineering Physics

Supervisors: Professor João Pedro Bettencourt de Melo MendesProfessor Pedro Miguel Félix Brogueira

Examination Committee

Chairperson: Professor Pedro Domingos Santos do SacramentoSupervisor: Professor João Pedro Bettencourt de Melo MendesMember of the Committee: Professor Luísa da Conceição dos Santos

de Canto e Castro de LouraMember of the Committee: Professor Rui Manuel Agostinho Dilão

November, 2014

Resumo

Durante of últimos anos o Ministério da Educação e Ciência (MEC) tem sido acusado de utilizar con-

tratos temporários em vez de permanentes para satisfazer as necessidades de professores do sistema

educativo. No fim de 2013 a Comissão Europeia exigiu que este tratamento discriminatório dos pro-

fessores a contrato temporário acabasse. Nós desenvolvemos um modelo de dinâmica de sistemas da

procura de professores para ajudar o MEC a estimar as necessidades de docentes do sistema, quanto

tempo esperamos que estas durem, e para fornecer ideias para o desenvolvimento de uma polı́tica de

contratação de professores.

Simulámos a evolução das necessidades de professores até 2063 utilizando as previsões da taxa

de natalidade do Instituto Nacional de Estatı́stica (INE). As polı́ticas de contratação testadas foram

não contratar, contratar professores para manter constante o rácio alunos por professor e contratar de

acordo com as necessidades de horas lectivas.

Concluı́mos que em cada momento (a) os professores devem ser contratados utilizando uma métrica

que reflita necessidades reais, por exemplo, necessidades de horas lectivas, e que (b) o número es-

timado de professores necessários pode ser totalmente contratado, porque as necessidades não vão

diminuir com o tempo.

Palavras-chave: Necessidades de Professores, Educação, Formulação de Polı́ticas, Dinâmicade Sistemas

iii

Abstract

For many years the Portuguese Ministry of Education and Science (MEC) has been accused of using

temporary contracts instead of permanent ones to satisfy the system’s needs for teachers. By the end of

2013 the European Commission demanded the termination of this discriminatory treatment of temporary

hired teachers. We developed a system dynamics model of teacher demand to help MEC estimate the

teacher needs for the education system and how long will these needs will last, and to provide insights

to the development of a teacher hiring policy.

We simulated the evolution of teacher needs until 2063 using birth rate predictions from National

Statistics Agency. The hiring policies tested for were no hiring, hiring to keep a constant student to

teacher ratio, and hiring according to teaching hours needs.

Our main conclusions were that in any given moment (a) teachers should be hired using some metric

for teacher needs, for example, total teaching hours, and (b) the estimated number of teachers needed

could actually be hired because these needs would not decrease with time.

Keywords: Teacher Demand, Education, Policy Design, System Dynamics

v

Preamble

A teacher demand model for basic education. This may seem an unusual subject for a thesis to obtain a

Master of Science Degree in Physics Engineering, and so I decided to start the work by presenting how

I got here.

I was first attracted to physics in high school because of its objectivity and its ability to predict the

future. During the degree I learned about many models that despite being approximations of reality were

able to generate very accurate and useful results.

Another area that peaked my interest was trying to predict human behavior. Inspired by the simula-

tions we did in physics I believed that it would be possible to simulate the whole world and everyone on it

(how naive). Through this ”world simulation” we would test our policies, and use the acquired knowledge

to design a better society. This encouraged me to start reading about agent based modeling.

I discovered that agent based models belonged to a broader class of models called microscale mod-

els, in which complex phenomena appear from the interaction of multiple agents moved by simple rules.

An alternative to microscale models were macroscale models, that usually use differential equations to

describe the behavior of variables. In these, complex behaviors are generated from simple equations for

the flows (variables’ rates of change).

Knowing how physics was developed, I decided to begin by learning about macroscale models.

After all, physics started by explaining phenomena at the scale to which they were observed, because

macroscale models tend to be simpler to develop and validate. Only if this explanation failed or became

too complex did we reduce the scale of analysis.

I wanted to learn a useful methodology, and knowing that those most in use today were agent based

modeling, discrete event simulation and system dynamics, I decided to choose this last one for my thesis

because it helped develop macroscale models.

All that I needed now was a problem. We should always aim at problems that are relevant for our

clients, which are the people whose actions we must influence for our work to have impact. Since I

considered that education was very important component of a functional society I decided to restrict my

universe of problems to educational ones. Assuming that the client would be the Portuguese Ministry of

Education and Science - Ministério da Educação e Ciência (MEC), I sent an email to the office of the

Minister of Education and Science (Appendix A).

After a few emails a meeting was arranged with minister’s office staff member Pedro Miguel da Rosa

Janeiro. In this meeting I was suggested to analyze which were the factors that affected a university’s

vii

students degree of employability. Although it was an educational problem, it was not one that directly

concerned MEC, and so we decided to adapt it. The adapted problem agreed with my supervisors was:

how should MEC use employability ratings to control the number of vacancies that universities could

open for each curriculum.

While analyzing this problem I attended a lecture at my university by João Oliveira Baptista, a former

Physics Engineering student who was the deputy director of General Office of Education and Science

Statistics - Direcção-Geral de Estatı́sticas da Educação e Ciência (DGEEC). In his lecture he presented

some statistics about the employability of students from several faculties. Given the proximity of the

subject to my work I decided to also send him an email, asking if we could arrange a meeting to discuss

some of my results (Appendix A).

In this meeting, the results and methodology were considered interesting, but this was still a problem

outside of the scope of MEC. By the end of the meeting we agreed that a real problem concerning MEC

was estimating how many teachers should be hired within each recruitment group, based on the needs

in the forthcoming years. Faced with an option to analyze a problem of real importance, I decided to

switch my dissertation topic to the creation of a teacher demand model.

We agreed that to be able to access the available data they would provide a desk and a computer

for me to work at DGEEC. After a few more emails (Appendix A) another meeting took place with the

head of the department in charge of the basic and secondary educational data. Then, I began working

at DGEEC.

The work that follows was a result of this rather unusual, but enjoyable, sequence of events.

viii

Contents

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1 Introduction 1

1.1 Problem Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Proposed Research Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Adapting the Research Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5.2 System Dynamics Applications to Education . . . . . . . . . . . . . . . . . . . . . 5

1.6 Work Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Model Structure and Development 7

2.1 Subsystem Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Student Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Changing Fail Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Failures in First Grade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.3 Dropouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.4 Age Dependant Fail Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Teaching Hours Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Teaching Hours for Lessons - Regular Education . . . . . . . . . . . . . . . . . . . 18

2.3.2 Other Needs for Teaching Hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Teaching Hours from Board Teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.1 Board Teacher Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.2 Other Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 QZP Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

ix

2.5.1 Student Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5.2 Teacher Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5.3 Number of Classes Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Policy Analysis 39

3.1 Policies to be Tested . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.1 Student to Teacher Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.2 Teaching Hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Student to Teacher Ratios vs Teaching Hours . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Teaching Hours Policy Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Scenario Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Conclusion 52

4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Evaluation of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Contribution to Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.1 Modes of Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3.2 Performance Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3.3 Leverage Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4 Limitations and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Bibliography 58

A Emails 59

A.0.1 Email for the office of the Minister of Education . . . . . . . . . . . . . . . . . . . . 59

A.1 Correspondence with Pedro Janeiro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

A.1.1 Before the Meeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

A.1.2 After the Meeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

A.2 Correspondence with João Baptista . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

A.2.1 Before the Meeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

A.2.2 After the Meeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

x

List of Tables

2.1 Maximum and minimum deviations for students in grades 1 to 4 for the basic student model. 10

2.2 Maximum and minimum deviations for students in grades 5 to 9 for the basic student model. 10

2.3 Dropout rates for students that failed in a given grade with a given age. The considered

age is at 31 of December in the school year after the one in which they failed. . . . . . . . 14

2.4 Maximum and minimum deviations for students in grades 1 to 4 for the final model. . . . . 17

2.5 Maximum and minimum deviations for students in grades 5 to 9 for the final model. . . . . 18

2.6 Percentage deviations between the results of the basic estimator and the measured data

for the first cycle (grades 1 to 4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19


for the second cycle (grades 5 to 6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19


for the third cycle (grades 7 to 9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.9 Variable names used in the equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.10 Percentage deviations between the results of the density function estimator and the mea-

sured data for the first cycle (grades 1 to 4). . . . . . . . . . . . . . . . . . . . . . . . . . . 21


sured data for the second cycle (grades 5 to 6). . . . . . . . . . . . . . . . . . . . . . . . . 21


sured data for the third cycle (grades 7 to 9). . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.13 Number of lecture hours needed by a 1st cycle class. . . . . . . . . . . . . . . . . . . . . 22

2.14 Number of lecture hours needed by a 2nd cycle class. . . . . . . . . . . . . . . . . . . . . 22

2.15 Number of lecture hours needed by a 3rd cycle class. . . . . . . . . . . . . . . . . . . . . 22

2.16 Data available for each teacher relative to the year 2013/2014. . . . . . . . . . . . . . . . 23

2.17 Fractions of the teaching hours that could be given that are actually given and that are

reduced from the teacher’s teaching hours (recruitment groups with single teacher). . . . 24

2.18 Fractions of the teaching hours that could be given that are actually given and that are

reduced from the teacher’s teaching hours. . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.19 Number of teachers of the groups 110 and 230 that terminated their contracts in 2013/2014. 33

3.1 Parameters that define the base scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

xi

3.2 Parameters that define the three scenarios that contemplate the model’s uncertainty in

parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

xii

List of Figures

1.1 Chain model of an educational system taken from Tavares (1995). . . . . . . . . . . . . . 3

2.1 Simple subsystem diagram showing how the different model components relate to one

another. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Stock and flow diagram reflecting the student flow through the grades. . . . . . . . . . . . 8

2.3 Stocks and flows representing the aging of children until their 6th birthday. . . . . . . . . . 9

2.4 Model results (blue) compared with the measured data (red) for the number of students

in each grade in the public education system. . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Model results (blue) compared with the measured data (red) for the number of students in

each grade in the public education system. This model considers the historical reduction

of fail rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12



of fail rates and failures for first grade students. . . . . . . . . . . . . . . . . . . . . . . . . 13



of fail rates and failures for first grade students. It also imposes the estimated number of

children with 5 years from Statistics Portugal - Instituto Nacional de Estatı́stica (INE). . . . 14


in each grade in the public education system. This model considers the historical reduc-

tion of fail rates, failures for first grade students and dropout rates. It also imposes the

estimated number of children with 5 years from INE. . . . . . . . . . . . . . . . . . . . . . 15



of fail rates and their dependency with age, failures for first grade students and dropout

rates. It also imposes the estimated number of children with 5 years from INE. . . . . . . 16



of fail rates and their dependency with age, failures for first grade students and dropout

rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

xiii

2.11 Histograms of students (of a year or cycle) in schools with a given number of students (of

that year or cycle) in 2013. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.12 Stock and flow diagram that models the aging and retirement of board teachers. . . . . . 27

2.13 Model results (blue) compared with the measured data (red) for the number of board

teachers of the recruitment group of 1st cycle teachers in the public education system. . . 28


teachers of the teaching specialty 2nd cycle Mathematics and Science in the public edu-

cation system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.15 Stock and flow diagram that models the aging and retirement (early or not) of board teach-

ers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29


teachers of the recruitment group of 1st cycle teachers in the public education system. It

was considered that 50% of teachers that became 55 years old retired and that 67% of

teachers that became 60 years old also retired. . . . . . . . . . . . . . . . . . . . . . . . . 29



cation system. It was considered that 15% of teachers that became 55 years old retired

and that 60% of teachers that became 60 years old also retired. . . . . . . . . . . . . . . 30


teachers of the recruitment group of 1st cycle teachers in the public education system.

The retirement age rose to 66 in 2014. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31



cation system. The retirement age rose to 66 in 2014. . . . . . . . . . . . . . . . . . . . . 31



The retirement age rose to 66 in 2014 and teacher recruitment programs were considered. 32



cation system. The retirement age rose to 66 in 2014 and teacher recruitment programs

were considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32



The retirement age rose to 66 in 2014. Teacher recruitment programs were considered

and so were the contract terminations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33


teachers of the teaching specialty 2nd cycle Mathematics and Science in the public ed-

ucation system. The retirement age rose to 66 in 2014. Teacher recruitment programs

were considered and so were the contract terminations. . . . . . . . . . . . . . . . . . . . 34

xiv


each grade in the public education system in Educational Zone Board - Quadro de Zona

Pedagógica (QZP) 7. This simulation considered FractionQZP7 = 0.267. . . . . . . . . . . 36


in each grade in the public education system in QZP 7. This simulation considered

FractionQZP7 = 0.2936. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36


in each grade in the public education system in QZP 7. This simulation considered

FractionQZP7 = 0.2936 and started in 2007. . . . . . . . . . . . . . . . . . . . . . . . . . . 37



cation system in QZP 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1 Model results for the supply and demand of teachers when no teachers are hired in the

base scenario for group 110. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Model results for the supply and demand of teachers when no teachers are hired in the

base scenario for group 230. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Model results for the expected number of hired teachers needed as a result of the different

policy alternatives in the base scenario for group 110. . . . . . . . . . . . . . . . . . . . . 42



3.5 Model results for the expected number of hired teachers needed as a result of the differ-

ent policy alternatives in the base scenario for group 110. A class size increase of two

students per class took place in 2040. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43





3.8 Model results for the expected number of hired teachers needed as a result of the differ-

ent policy alternatives in the base scenario for group 110. A class size increase of two

students per class took place in 2040. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.9 Model results for the expected number of hired teachers needed for group 110 in different

scenarios when no teachers are hired. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46


scenarios when no teachers are hired. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46


scenarios when 100% of the teaching hours needs are hired. . . . . . . . . . . . . . . . . 47


scenarios when 100% of the teaching hours needs are hired. . . . . . . . . . . . . . . . . 47

xv

3.13 Model results for the board teachers hired of group 110 in different scenarios when 100%

of the teaching hours needs are hired. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.14 Model results for the board teachers hired of group 230 in different scenarios when 100%

of the teaching hours needs are hired. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.15 Model results for the board teachers of group 110 in each age group. . . . . . . . . . . . . 49

3.16 Model results for the board teachers of group 230 in each age group. . . . . . . . . . . . . 49

3.17 Model results for the board teachers of group 110 in each age group with a class size

reduction of 4 in 2020. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.18 Model results for the board teachers of group 230 in each age group with a class size

reduction of 4 in 2020. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

xvi

Acronyms

DGEEC General Office of Education and Science Statistics - Direcção-Geral de Estatı́sticas da Educação

e Ciência. viii, 3, 26, 35

GEP Planning and Studies Office - Gabinete de Estudos e Planeamento. 10

INE Statistics Portugal - Instituto Nacional de Estatı́stica. xiii, 9, 13, 14, 16, 17, 35, 40, 45

MEC Ministry of Education and Science - Ministério da Educação e Ciência. vii, viii, 1

. 67

QZP Educational Zone Board - Quadro de Zona Pedagógica. xv, 7, 17, 35–38, 53, 55

. 67

xvii

Chapter 1

Introduction

1.1 Problem Context

Historically Portugal has always had a very centralized education system, possibly because it is a rel-

atively small country without substantial regional differences. One of the centralized components is a

unified teacher hiring process, which is used to guarantee that the best teachers are hired, and only in

the amount necessary to satisfy the system’s need for teachers.

In Portugal there are two main types of teachers: board teachers and hired teachers. Board teachers

have a ”permanent” contract that is very difficult to terminate and hired teachers only have one-year

contracts, that may or may not be renewed depending on whether they are still needed. The underlying

logic is that there are permanent system needs, which last for a reasonable amount of time and justify

the hiring of a board teacher; and temporary system needs, which have a short time duration before

they disappear and so can be satisfied by a hired teacher.

For several years, Ministry of Education and Science - Ministério da Educação e Ciência (MEC)

has been accused of using temporary hired teachers to satisfy permanent system needs (de Notı́cias,

2014). In the end of 2013 the European Commission demanded that the hired teachers that have al-

ready worked in public education for several years should join the board, and Ministry of Education and

Science - Ministério da Educação e Ciência (MEC) proposed a semi-automatic hiring process for teach-

ers (Educare, 2014a). Besides these demands the national association of hired teachers (Associação

Nacional de Professores Contratados) is also going to file a complaint to the European Commission to

know the criteria for the number of vacancies that were opened for new board teachers and about their

distribution among the various subjects (Educare, 2014b).

1.2 Proposed Research Question

Justifying their teacher hiring processes is nowadays an important problem for MEC. As described in

chapter this problem was first voiced by the vice director of General Office of Education and Science

Statistics - Direcção-Geral de Estatı́sticas da Educação e Ciência (DGEEC). The implied research ques-

1

tion was:

• What will be the need for teachers in the coming years?

This research question was decomposed into three sub-questions:

• How many students are expected to be enrolled in each grade in the public education system?

• How many teachers will be needed to teach those students?

• How many board teachers will we have over time (under several hiring policies)?

1.3 Literature Review

Similar goals have also been pursued in other countries, and a variety of methods were used to estimate

how many teachers were needed, either for recruitment or for training purposes.

In the United Kingdom a planning effort has been made with respect to how many teachers should

be trained, starting with a report in 1990 (DES, 1990). Two more technical descriptions of the model

were published in 1998 (DfEE and GSS, 1998) and in 2013 (DfE, 2013). Their demand forecasts were

made using student to teacher ratios individualized for different subjects. Early retirement rates were

also estimated by age, gender and school level (primary or secondary) using time series forecasts.

Predictions regarding the number of elementary and secondary teachers were also made in the USA

(NCES, 2014). Their model predicted the future demand for teachers using linear regression to estimate

the future evolution of student to teacher ratios.

In Australia several governmental studies have been published from 1998 (MCEECDYA, 1998) to

2004 (MCEETYA, 2004) which estimated teacher supply and demand by using student enrollment levels

and student to teacher ratios. They presented aggregate national data for the total number of teachers,

but not for each recruitment group independently. Estimations of the demand generated from retirements

were also made for the total teacher workforce.

Approximately 20 years ago, Portugal had one of the most developed planning models for education.

LINSSE, also called chain model of an educational system (GEP, 1992), was an educational planning

model developed by Professor Luı́s Valadares Tavares while he was director of the statistics department

of the Portuguese Ministry of Education. It is based on a student model which consists in a matrix

through which students flow as years advance. Its representation is depicted in figure 1.1.

The model describes that by the end of each school year students will either pass to the next grade

or they will remain in the same grade. For both cases we can have a drop-out flow of students that leave

the education system.

The LINSSE model also made predictions regarding the number of classes, number of teachers

and number of classrooms required. It was developed and used when the mandatory school age was

increased so that the Portuguese government might be able to plan adequately.

2

Figure 1.1: Chain model of an educational system taken from Tavares (1995).

1.4 Adapting the Research Question

All the previously mentioned models aimed at obtaining predictions for certain variables. Decision-

makers will plan based on these predictions and develop policies which best fit their goals. But should

predictions really be the goals of modeling?

In a paper by Coyle (1978) he discusses why this ”naive planning” fails. He states that forecasts

fail not only because of unpredictable events, which were not considered in the plan, but also because

we make predictions wrong by responding to them. If there are predictions that unemployment will rise

governments will act to prevent it (successfully or not) and invalidate the prediction itself.

During the first meeting with the direction of General Office of Education and Science Statistics -

Direcção-Geral de Estatı́sticas da Educação e Ciência (DGEEC), one thing they told me was that all

of my estimations could become completely wrong if new laws were approved. Any change in system

parameters such as the class size, number of teaching hours required, or any other would eliminate the

validity of the predictions.

To prevent this issue we have decided to adapt our research question to one that had a robust

answer:

• What teacher hiring policy should be applied to minimize the teacher deficit while guaranteeing

that no teachers were hired and later ceased to be needed?

But what is the difference between the two? If policies are based on model predictions then why is

this a better research question?

To address this issue let us distinguish a policy from a decision. Forrester (1992) defines both as:

[...] ”policy” is a rule that states how day-by-day operating decisions are made. ”Deci-

sions” are actions taken at any particular time and result from applying policy rules to partic-

ular conditions that prevail at the moment.

3

As stated, decisions come from the application of policies (policy rules). For example, a teacher hiring

decision would state in a specific moment how many teachers should be hired for each recruitment

group. A teacher hiring policy would state how many teachers should be hired at any moment as a

function of the available information at that time.

This definition of policy originates from control theory. In control engineering, no person is respon-

sible for constantly regulating a system. For example, when a rocket went to the moon, it had sensors

to measure whether it was on the right trajectory or deviating from it. These measurements were not

interpreted by a human that would correct the rocket’s trajectory. What control engineers do is to design

a controller that automatically decides which corrections are necessary as a function of the available

information.

Considering the above definition of policy, this second research question becomes one of feedback

controller design. None of the traditional econometric models allow these kinds of analysis because

to design a feedback controller we need a causal model of the system to estimate its behavior in new

situations. This limitation of econometric models is stated by Sterman (1991):

[...] econometrics fails to distinguish between correlations and causal relationships. Sim-

ulation models must portray the causal relationships in a system if they are to mimic its

behavior, especially its behavior in new situations. But the statistical techniques used to es-

timate parameters in econometric models don’t prove whether a relationship is causal. They

only reveal the degree of past correlation between the variables, and these correlations may

change or shift as the system evolves.

Due to these drawbacks other tools were needed to apply control theory in management, and these

applications founded the field of System Dynamics which will be the methodology used in this work.

1.5 System Dynamics

1.5.1 History

System Dynamics was developed by Jay Wright Forrester inspired by a discussion of irregular hiring

patterns at General Electric (Forrester, 2007):

They were puzzled by why their household appliance plants in Kentucky were sometimes

working three and four shifts and then, a few years later, half the people would be laid off. It

was easy enough to say that business cycles caused fluctuating demand, but that explanation

was not convincing as the entire reason. After talking with them about how they made hiring

and inventory decisions, I started to do some simulation. [...] It became evident that there

was potential for an oscillatory or unstable system whose behavior was entirely internally

determined. Even with constant incoming orders, one would get employment instability as a

consequence of commonly used decision-making policies within the supply chain. That first

4

inventory-control system with pencil and paper simulation was the beginning of the system

dynamics field.

Contrary to the common belief the undesirable behavior watched was not a consequence of external

factors, outside management control, but of hiring policies that were badly designed. A compilation of

the results obtained through these dynamical models was published in Industrial Dynamics (Forrester,

1961) which launched the field of System Dynamics.

Another early application of System Dynamics was Urban Dynamics (Forrester, 1969), which showed

that major urban policies such as building low-cost housing actually created poverty instead of solving

it. This occurs because these houses attracted people that needed jobs while simultaneously occupying

space where jobs could be created.

A following high impact work in the field was World Dynamics (Forrester, 1971) and its successor

Limits to Growth (Meadows, 1972). These projects analyzed the interactions between the world econ-

omy, population and ecology from a system’s viewpoint. They predicted that if the present growing trends

didn’t change our planet would reach its limit within the next 100 years.

A number of other early system dynamics studies are reviewed by Coyle (1978), which include work

in the chemical plant construction industry, the U.K. paper industry and the mining industry among

others.

1.5.2 System Dynamics Applications to Education

The first educational application published was by Chen et al. (1981). In it they discovered that policies

aimed at equalizing educational expenditures in city school districts were only effective for one or two

years, and had their effect diminished or vanished in the long run.

System Dynamics was also applied to model the higher education and professional labor force re-

quirements in Australia (Galbraith, 1984) and later to study the impact of government incentives on

competition between faculties and schools (Galbraith, 1998). Many other educational applications of

system dynamics are reviewed by Kennedy (2011).

Galbraith (1999) also made a study of the supply and demand of teachers in Australia, in which he

reinforced that predictions shouldn’t be the goal:

Prediction at a level of precision, given the plethora of variables is beyond the capabilities

of any forecasting tool. Further a confluence of social and economic coincidences at any

point in time could give unwarranted credence to a flawed formula, with dire future conse-

quences. But this does not mean that efforts to understand and address the supply-demand

problem analytically are futile [...] Rather than aim to meet hypothetically determined specific

output levels, efforts should be focused on the rates of change that govern the levels - with

the aim of stabilizing variations in the system as a whole, so enabling emerging discrepancies

to be addressed speedily.

5

1.6 Work Plan

Sterman (2000) states that to build a successful system dynamics model our work should include all of

these activities:

• Problem Articulation

• Formulation of a Dynamic Hypothesis

• Formulation of a Simulation Model

• Testing

• Policy Design and Evaluation

These activities can be performed several times, and iterations between them are advisable to obtain

a good model.

The problem articulation has already been done in this chapter, by adapting the proposed research

question to one that is robust to changes in the system.

During the following chapter we will develop a student model, an estimator for the needs for teach-

ers associated with those students, and a teacher model. For all of them we will iterate a few times

between formulation of dynamic hypothesis, turning these into simulation models and testing them. The

reformulation of hypothesis will be made taking into consideration the model results.

Finally in the last chapter we will design policies to address the teacher hiring problem, and evaluate

their relative performance.

6

Chapter 2

Model Structure and Development

This chapter discusses the development of the teacher demand model. We start by presenting an

overview of the model in a diagram. Each model component is then developed by starting with a basic

version of it and increasing its complexity iteratively, by analyzing the differences between the model

results and the observed behaviors. Although point predictions are not the goal, we have decided to

calibrate the model to approximately reproduce historical behavior to increase our confidence in it.

The development of the model proceeds as follows: first, the results from the student enrollment help

estimate the demand for teachers, which are based on teaching hours needed. Next, the board teacher

model, which reflects the aging and retirement of the board teacher workforce, is used to estimate the

teaching hours offered by board teachers. Finally, in the last section the model is adapted to reflect just

one QZP.

2.1 Subsystem Diagram

To obtain a general view of the model components and their relations we present a simple diagram in

figure 2.1. Blue circles represent model components and the remaining variables outside of the circles

are exogenous to the model. All of them except the Birth Rate (number of births per year) are controlled

by policy options and their values were either based on historical data or on the legislation itself.

The birth rate is used as input to the Student Model, which predicts how many students are enrolled

in each grade over time. Using information about the legislated class size and the number of students of

the schools we estimate the number of classes accommodating these students. Knowing the teaching

hours needed per class and the hours required for other activities performed by teachers we estimate

the Teaching Hours Needs.

The Board Teacher Model predicts the number of board teachers over time, using the number of

newly hired teachers and the teacher retirements (which depend on the hiring policy and the retirement

legislation respectively). We obtain the Teaching Hours from Board Teachers by knowing the number of

teaching hours each board teacher provides (stated in the legislation).

The difference between the teaching hours needs and the teaching hours from board teachers will

7

Figure 2.1: Simple subsystem diagram showing how the different model components relate to oneanother.

be the unfulfilled teaching hours (or excessive if we have too many teachers). The goal of my work is to

test different Board Teacher Hiring Policies and observe whether these generate unfulfilled or excessive

teachers for several external conditions.

2.2 Student Model

To predict the number of students enrolled in each grade in the public education system we start by

considering the flow of students through consecutive stocks as years advance. Once a school year is

over students either advance to the next grade or remain in the same grade but one year older. The

model schematics is displayed in figure 2.2.

Figure 2.2: Stock and flow diagram reflecting the student flow through the grades.

The general format of the outflow equations is:

8

outflow1(t) = fraction1 (inflow1(t− 1) + inflow2(t− 1)) (2.1)

This equation states that the outflow in a given time t is the sum of the inflows delayed by one year

multiplied by a fraction - the pass (or fail) fraction. The general equations are:

Pass Flow i to i+ 1 with j failures(t) = Pass Fraction i to i+ 1

∗ [Pass Flow i− 1 to i with j failures(t− 1) + Failure Flow i to i+ 1 with j-1 failures(t− 1)]

Failure Flow i to i+ 1 with j failures(t) = (1− Pass Fraction i to i+ 1)

∗ [Pass Flow i− 1 to i with j failures(t− 1) + Failure Flow i to i+ 1 with j-1 failures(t− 1)]

(2.2)

The student flow entering the first grade is determined by a sequence of stocks depicting a child’s

aging from birth to their 6th birthday. The model schematics is displayed in figure 2.3.

Figure 2.3: Stocks and flows representing the aging of children until their 6th birthday.

These outflows are given by the inflow delayed by one year (outflow(t) = inflow(t− 1)).

The birth rate is an exogenous variable imposed using the data from INE.

Since the purpose of the model is to estimate the teacher hiring needs for the public education

system we need to restrict the student population to the students that enroll in it. We considered that

there were no transferred students between education systems, and so students only entered the public

education system in the first grade. The fraction of students enrolling in public education was considered

constant for the duration of the simulation and equal to its value at the starting date.

This model has the following assumptions:

• Constant fail rates

• Absence of failures in first grade

• Absence of dropouts

• Fail rates independent of the previous number of failures

• Absence of deaths, immigration or emigration

• All students that enter first grade are 6 years old

• Absence of students transferring between education systems (ex: public to private)

• Constant fraction of students entering each education system

• After 5 failures through the 9 grades students abandon the system

9

The model developed so far is equivalent to the model LINSSE (GEP, 1992) developed by Professor

Luı́s Valadares Tavares e by the Planning and Studies Office - Gabinete de Estudos e Planeamento

(GEP). The model results are displayed in figure 2.4.

Figure 2.4: Model results (blue) compared with the measured data (red) for the number of students ineach grade in the public education system.

We notice visible deviations between both datasets. Among them there is an excessive amount of

students through all grades for the final years of the simulation and a lack of first grade student in the

first years. Since the scale is amplified in the figures the percentage deviations are shown in tables 2.1

and 2.2.

Students 1st Grade Students 2nd Grade Students 3rd Grade Students 4th Grade

Max Deviation 3.4% 14.3% 13.8% 12.0%Min Deviation -7.1% -5.0% -3.9% -4.6%

Table 2.1: Maximum and minimum deviations for students in grades 1 to 4 for the basic student model.

Students 5th Grade Students 6th Grade Students 7th Grade Students 8th Grade Students 9th Grade

Max Deviation 11.0% 7.5% 14.5% 29.6% 35.9%Min Deviation -3.3% -0.8% -0.3% -0.2% -0.2%

Table 2.2: Maximum and minimum deviations for students in grades 5 to 9 for the basic student model.

We notice an excessive number of students of around 10% in grades 2 to 6 and from 15% to 35%

in grades 7 to 9. Since the deviations grow in the more advanced grades we expect to be missing two

different phenomena. The first will affect all grades and the second will be specific to the 3rd cycle

10

(grades 7 to 9) with a greater effect on grades 8 and 9. The lack of students in first grade also indicates

a third phenomena.

In the following sections these phenomena are included in the model.

2.2.1 Changing Fail Rates

The first deviation observed is an excessive amount of students in almost all grades. A possible expla-

nation the model needs to replicate is a reduction of fail rates. If less students fail then the total number

of students enrolled declines.

Over the last 15 years failing students started to be used only as a last resort and discouraged by the

Ministry of Education and Science. By considering the historical pass rates we modified the model to

make the pass rates grow and stabilize on their present values (following an S-shaped curve associated

with the diffusion of ideas).

The following assumptions are still valid:

• Constant fail rates (from 2013 onward)

• Absence of failures in first grade








Since it is very hard to estimate how will the fail rates evolve we have estimated them to be constant

from 2013 onward. This is then a model improvement that does not improve future predictions, but to

exclude it could generate a bias in other phenomena.

The model results are displayed in figure 2.5.

It is now noticeable that the student excess in grades 1 to 6 was greatly reduced, and so was part of

the excess in grades 7 to 9. We will now try to explain the lack of first grade students in the first years of

the dataset by introducing failures in the first grade.

2.2.2 Failures in First Grade

Although according to the law no students should fail in first grade this rule wasn’t always applied. We will

then introduce a growing pass rate for first grade students from 0.93 in 2001 to 0.99 in 2013 (according

to the same behavior as the remaining pass rates).

The following assumptions still apply:

11

Figure 2.5: Model results (blue) compared with the measured data (red) for the number of students ineach grade in the public education system. This model considers the historical reduction of fail rates.









The obtained results are presented in figure 2.6.

We noticed that a large portion of the first grade student excess was explained but we still have an

excessive number of students in the most recent years. A possible explanation the emigration that has

been taking place due to the economic crisis. To test that theory we imposed the estimated number of

children with 5 years from INE. We are then, implicitly, considering deaths, immigration and emigration

that take place during these years.

The following assumptions are then remaining:


12

Figure 2.6: Model results (blue) compared with the measured data (red) for the number of students ineach grade in the public education system. This model considers the historical reduction of fail ratesand failures for first grade students.



• Absence of deaths, immigration or emigration for ages above 5 years





The obtained results are displayed in figure 2.7.

The number of children enrolled in first grade is now reasonably accurate. A likely explanation for the

previously observed deviations is then the deaths, emigration or immigration of children under 5 years.

We still lack some students in grades 2 to 6 that seem to move, which indicates a cohort of students

that entered 1st or second grade in years 2001 to 2007, possibly immigrants of ages above 5 that were

enrolled in these grades. This phenomena was excluded because it would not bias other parameter

estimation or improve the model’s future predictions.

A large excess of students in grades 7 to 9 still exists. We will now test whether these can be

explained by students that drop out of school.

13

Figure 2.7: Model results (blue) compared with the measured data (red) for the number of students ineach grade in the public education system. This model considers the historical reduction of fail ratesand failures for first grade students. It also imposes the estimated number of children with 5 years fromINE.

2.2.3 Dropouts

Until 2010 students with 15 years of age or older could stop attending school. From 2010 onward

the minimum age to drop out of school became 18 years there are alternatives to regular education

for children that have difficulty completing it. We will assume that the dropout fractions remained the

same before and after this legislation, but after it all dropout students younger than 18 changed to other

education routes.

The considered dropout rates obtained through an adjustment of the model results to the historical

data are displayed in table 2.3. For this adjustment we imposed the number of 5 year old children to

reduce the distortions coming from the emigration and immigration.

Grade

Age 7th 8th 9th15 80% 80% 35%16 100% 80% 50%17 100% 70%18 90%

Table 2.3: Dropout rates for students that failed in a given grade with a given age. The considered ageis at 31 of December in the school year after the one in which they failed.

The following assumptions remain:


14








Figure 2.8: Model results (blue) compared with the measured data (red) for the number of students ineach grade in the public education system. This model considers the historical reduction of fail rates,failures for first grade students and dropout rates. It also imposes the estimated number of children with5 years from INE.

There is still an excessive number of students in grades 7 to 9 so we decided to see the behavior of

the number of enrolled students discriminated by age. Since these graphs had substantial differences

for all grades we decided to remove the assumption that fail rates are independent of age.

2.2.4 Age Dependant Fail Rates

We only had the pass rates data discriminated by age in the most recent years so it was assumed that

these scaled with age in the same proportion for the older years of the dataset.

The following assumptions still remained:

15








Figure 2.9: Model results (blue) compared with the measured data (red) for the number of students ineach grade in the public education system. This model considers the historical reduction of fail ratesand their dependency with age, failures for first grade students and dropout rates. It also imposes theestimated number of children with 5 years from INE.

The remaining deviations between the model and the historical data now appear reasonably small

for all grades.

2.2.5 Conclusion

Although emigration and immigration are still not included in the model we decided to keep the system-

atic error generated by neglecting them. By using the birth rate instead of the number of 5 year old

children as an exogenous variable we can then increase the time horizon of the model without having to

include births or population forecasts.

16

The final assumptions are then:








Figure 2.10: Model results (blue) compared with the measured data (red) for the number of students ineach grade in the public education system. This model considers the historical reduction of fail ratesand their dependency with age, failures for first grade students and dropout rates.

To evaluate the model’s accuracy the percentage deviations are displayed in tables 2.4 and 2.5.

Students 1st Grade Students 2nd Grade Students 3rd Grade Students 4th Grade

Max Deviation 4.5% 4.9% 7.3% 4.5%Min Deviation -2.2% -5.9% -4.0% -5.5%

Table 2.4: Maximum and minimum deviations for students in grades 1 to 4 for the final model.

Since these were all bellow 10 % and the majority were bellow 5% we found the model’s accuracy

reasonable for the purpose.

17

Students 5th Grade Students 6th Grade Students 7th Grade Students 8th Grade Students 9th Grade

Max Deviation 4.3% 4.5% 5.4% 2.3% 2.8%Min Deviation -2.5% -0.8% -1.1% -6.6% -5.0%

Table 2.5: Maximum and minimum deviations for students in grades 5 to 9 for the final model.

2.3 Teaching Hours Needs

After having the student enrollment forecasts we can estimate the needs for teachers.

Through this work we consider needs for teachers as needs for teaching hours. This was decided

because if a board teacher doesn’t have enough teaching hours assigned to him, then he can be moved

to another school in which he is needed. By using teaching hours aggregate at a national level (or at

a QZP level) to estimate the demand for new teachers we are making sure that no teachers are hired

for hours if we already have teachers that aren’t teaching enough hours in other schools. We are then

preventing the hiring of too many teachers by choosing this estimator of needs.

The teaching hours considered are the hours of actual classes and all other hours equivalent to them

(such as providing support to students). Activities that allow teachers to reduce the number of hours

they need to teach are also considered as teaching hours needs. This includes all reductions to the

teaching hours of teachers excluding age related reductions, which are considered as a reduction in the

number of hours board teachers provide.

2.3.1 Teaching Hours for Lessons - Regular Education

To know the teaching hours needed for (regular eduction) lessons we use the number of classes. For

that we developed an estimator for the number of classes of students in a given grade as a function of

the number of students in that grade (determined in the previous section).

For the number of classes only a few reliable numbers exist that can be used to test our estimators.

These are the number of classes in the 1st, 2nd and 3rd cycles (for the public education system) in the

years 2010/2011, 2011/2012. We also treated the data for the 2nd and 3rd cycles in the year 2013/2014.

We evaluated the estimator’s accuracy by applying it to the historical data of 2010/2011 and 2011/2012

but since this data was still not validated for the year 2013/2014 the model predictions were used instead.

Dividing by Class Size

The first estimator tested was a division by the class size that is stated in the law. This was also the

estimator used in the LINSSE model developed by GEP (GEP, 1992).

The legislation applied in the years 2010/2011, 2011/2012 and 2012/2013 was that the first cycle

classes could contain students for more than one grade. Classes should have 24 students if there was

only one grade in the classroom, 22 if there were more than one grade but in separate classrooms

and 18 if the several grades were all in the same place. In grades 5 to 12 classes should have a size

between 24 and 28 students (Ministério da Educação, 2007). In 2013/2014 the legislation changed and

the normal class size rose to 26 in the first cycle, and in grades 5 to 9 it is now between 26 and 30

18

students (Ministério da Educação e Ciência, 2013).

The number of classes obtained with this estimator and the percentage deviations to the measured

data are displayed in tables 2.6, 2.7 and 2.8. Since the estimator’s results are dependent on the class

size (cs) several options are presented.

Year Measured 1st cycle Div cs = 18 Div cs = 22 Div cs = 24 ∆% cs = 18 ∆% cs = 22 ∆% cs = 24

2011 19455 21321 17444 15990 10% -10% -18%2012 18807 20878 17082 15659 11% -9% -17%

Table 2.6: Percentage deviations between the results of the basic estimator and the measured data forthe first cycle (grades 1 to 4).

Year Measured 2nd cycle Div cs = 24 Div cs = 26 Div cs = 28 ∆% cs = 24 ∆% cs = 26 ∆% cs = 28

2011 9365 8768 8094 7516 -6% -14% -20%2012 9171 8513 7859 7297 -7% -14% -20%

2014 8734 8117 7510 6973 -7% -14% -20%

Table 2.7: Percentage deviations between the results of the basic estimator and the measured data forthe second cycle (grades 5 to 6).

Year Measured 3rd cycle Div cs = 24 Div cs = 26 Div cs = 28 ∆% cs = 24 ∆% cs = 26 ∆% cs = 28

2011 12610 11762 10857 10082 -7% -14% -20%2012 12851 11883 10969 10185 -8% -15% -21%

2014 12543 11955 11302 10495 -5% -10% -16%

Table 2.8: Percentage deviations between the results of the basic estimator and the measured data forthe third cycle (grades 7 to 9).

As we can see this estimator always generates a considerable systematic error except for cs = 18 in

the first cycle.

Student Distribution Function

The previously presented estimator assumed, implicitly, that the extra students from all schools that

didn’t fill a class could be merged to create classes with students from multiple schools. Since each

school organizes its students independently this assumption was not valid and can be the cause of the

observed deviations.

To account for that effect we developed an estimator in which each school has to have an integer

number number of classes that contain all of its students.

This estimator uses the student distribution function through schools of different sizes (size being the

number of students per school). The distribution of the total number of students in schools with a given

number of students is displayed in figure 2.11. First cycle students were grouped together because they

can also be grouped in classes of multiple grades.

The variable names used in the equations to come are displayed in table 2.9.

The estimator calculates for each school ”size” (number of students in a given cycle or grade in that

school) how many classes are needed. This number is rounded up and multiplied by the number of

schools with that size. The total number of classes is obtained by summing these results for all school

sizes:

19

Figure 2.11: Histograms of students (of a year or cycle) in schools with a given number of students (ofthat year or cycle) in 2013.

NC Number of ClassesNS/Sc Number of Students per School

NS(NS/S) Number of students in schools with NS/Sc students per schoolNSc(NS/Sc) Number of schools with NS/Sc students per school

cs Number of students per class

Table 2.9: Variable names used in the equations.

NC =∑

NS/Sc

ceil

(NS/Sc

cs

)NSc(NS/Sc) (2.3)

The number of schools with NS/Sc students (or a given year or cycle) per school is obtained by

dividing the total number of students (of that year or cycle) in schools with a given size by the number of

schools with that size:

20

NSc(NS/Sc) =NS(NS/S)

NS/Sc(2.4)

Combining the above equations we obtain:

NC =∑

NS/Sc

ceil

(NS/Sc

cs

)NS(NS/S)

NS/Sc(2.5)

To make the estimator explicitly dependent on the total number of students NS we used a change of

variables given by:

z =NS/SNS

f(z) =NS(NS/S)

NS

(2.6)

The variable z represents the fraction of the total student population served by a school of a given size

and the function f(z) represents the probability density function of students. By applying this change of

variables to equation 2.5 we obtain:

NC =∑z

ceil

(z NScs

)f(z)NSz NS

⇔

NC =∑z

ceil

(z NScs

)f(z)

z

(2.7)

The obtained results for this estimator are displayed in tables 2.10, 2.11 and 2.12.

Year Measured 1st cycle Func cs = 18 Func cs = 22 Func cs = 24 ∆% cs = 18 ∆% cs = 22 ∆% cs = 24

2011 19455 23397 19593 18030 20% 1% -7%2012 18807 22508 19229 17794 20% 2% -5%

Table 2.10: Percentage deviations between the results of the density function estimator and the mea-sured data for the first cycle (grades 1 to 4).

Year Measured 2nd cycle Func cs = 24 Func cs = 26 Func cs = 28 ∆% cs = 24 ∆% cs = 26 ∆% cs = 28

2011 9365 9626 8937 8343 3% -5% -11%2012 9171 9393 8650 8187 2% -6% -11%

2014 8734 9037 8344 7763 3% -4% -11%

Table 2.11: Percentage deviations between the results of the density function estimator and the mea-sured data for the second cycle (grades 5 to 6).

Year Measured 3rd cycle Func cs = 24 Func cs = 26 Func cs = 28 ∆% cs = 24 ∆% cs = 26 ∆% cs = 28

2011 12610 13319 12429 11521 6% -1% -9%2012 12851 13580 12477 11734 6% -3% -9%

2014 12543 13592 12703 12150 8% 1% -3%

Table 2.12: Percentage deviations between the results of the density function estimator and the mea-sured data for the third cycle (grades 7 to 9).

It is noticeable that this estimator is much more accurate than the previous one. To maximize it’s

accuracy we considered a class size of cs = 22 for the 1st cycle, cs = 24 for the 2nd cycle and cs = 26

21

for the third cycle.

A more detailed study would be needed to evaluate the impact of the legislative changes in the class

sizes that took place in 2013/2014. To gain perspective on the impact those changes might have in the

teacher hiring needs we considered in the base scenario that the values of cs remain the same (which

doesn’t seem to generate a substantial error considering the previous results), and then we changed

them in other scenarios.

Teaching Hours per Class

To determine the total number of teaching hours (for lessons) needed we need to know how many

teaching hours are required by each subject.

The teaching hours per subject defined in legislation (Ministério da Educação e Ciência, 2012) are

grouped in subject classes, and schools are free to distribute those hours for the different subjects.

We considered for this model that schools keep the more traditional distribution of hours, prioritizing

Portuguese and Mathematics and valuing all others equally. If the previously mentioned criteria is not

sufficient to distribute the hours then the first subjects mentioned in the tables of annex II of Decreto-Lei

n.o 139/2012 de 5 de Julho (Ministério da Educação e Ciência, 2012) are given a slight priority.

Through the application of these criteria we obtain the lecture hours per week displayed in tables

2.13, 2.14 and 2.15.

1st Cycle Number of Lecture Hours

1st Cycle Teacher Needs 25

Table 2.13: Number of lecture hours needed by a 1st cycle class.

2nd Cycle Number of Lecture Hours

Portuguese 6English 3History and Geography of Portugal 3Mathematics 6Natural Sciences 3Visual Education 2Technological Education 2Musical Education 2Physical Education 3Moral and Religious Education 1

Table 2.14: Number of lecture hours needed by a 2nd cycle class.

Number of Lecture Hours3rd Cycle 7th Grade 8th Grade 9th Grade

Portuguese 5 5 5English 3 3 3Foreign Language II 3 2 2History 3 3 3Geography 2 2 3Mathematics 5 5 5Natural Sciences 3 3 3Physics and Chemistry 3 3 3Visual Education 2 2 2Information and Communication Technologies 2 2 1Physical Education 3 3 3Moral and Religious Education 1 1 1

Table 2.15: Number of lecture hours needed by a 3rd cycle class.

22

To know the number of teaching hours required for lessons of each subject we can now multiply the

number of classes by the teaching hours per class in tables 2.13, 2.14 and 2.15.

We will now focus on the remaining needs for teaching hours.

2.3.2 Other Needs for Teaching Hours

To understand the remaining needs for teaching hours we used data that discriminated in which types

of activities these were used, which was only available in 2013/2014. In this data we ignored teachers

that were marked as being Senior Technician with Teacher Contract (Técnico Superior com Contrato

de Docente) and the ones that had specific situations stated (such as waiting for retirement, maternity

leave, etc.) The relevant data available for each individual teacher is shown in table 2.16.

Paid Hours Number of Teaching Hours for which the Teacher is PaidCycle Hours Number of Teaching Hours Associated with the teacher’s cycleTotal Reduction Total reduction of Teaching hours the teacher hasTotal Teaching Hours Sum of all activities equivalent to teaching hoursTeaching Hours Regular Teaching Hours in lessons to students in regular educationHours Support Hours giving support to studentsHours Support Teachers Hours giving assistance to other teachers in their lessonsTeaching Hours Non-Regular Teaching Hours in lessons to students that are not in regular educationHours Complementary Activities Teaching Hours giving complementary activities to students

Table 2.16: Data available for each teacher relative to the year 2013/2014.

To isolate the age related reductions we calculated, assuming that the only criteria for the reduction

of teaching hours was the age (ignoring years of service), the age reduction in lecture hours (Age

Reduction).

The other reductions to teaching hours (Other Reductions) were calculated by the following equation:

Other Reductions = max(Total Reduction− Age Reduction, 0) (2.8)

The maximum with zero was used to prevent negative values of Other Reductions for teachers that

don’t use all the age reductions to their teaching hours.

We calculate the expected teaching hours considering the reductions (Expected Teaching Hours):

Expected Teaching Hours = Paid Hours− Total Reduction (2.9)

Since the values in Expected Teaching Hours sometimes differed from the ones in Total Teaching

Hours we considered their difference to be an unjustified reduction in teaching hours (Unjustified Re-

ductions):

Unjustified Reductions = Expected Teaching Hours− Total Teaching Hours (2.10)

We then obtain the the following expressions for the Total Teaching Hours and for the Total Reduction:

23

Total Teaching Hours = Teaching Hours Regular + Hours Support + Hours Support Teachers

+ Teaching Hours Non-Regular + Hours Complementary Activities(2.11)

Total Reduction = Age Reduction + Other Reductions + Unjustified Reductions (2.12)

We also decided to evaluate the distribution of the possible reduction hours by the recruitment groups.

For that we calculated the maximum possible hours that teachers could give in lessons, which are

Paid Hours− Age Reduction. We then calculated the following fractions:

Fraction Effective Teaching Hours =Total Teaching Hours− Hours Complementary Activities

Paid Hours - Age Reduction(2.13)

Fraction Reductions =Other Reductions + Unjustified Reductions

Paid Hours - Age Reduction(2.14)

The results for the recruitment groups that have a single teacher are presented separately from the

remaining ones because for them the reductions in teaching hours make being a single teacher more

complicated (another teacher starts being needed to cover the reduction hours). The obtained fractions

are displayed in tables 2.17 and 2.18.

Recruitment Group Fraction Effective Teaching Hours Fraction Reductions

100 - Early Childhood Education 94.7% 5.5%110 - 1st Cycle of Basic Education 94.0% 6.1%Average 94.2% 6.0%

Table 2.17: Fractions of the teaching hours that could be given that are actually given and that arereduced from the teacher’s teaching hours (recruitment groups with single teacher).

We can see there are substantial differences in the amount of reduction to teaching hours that the

different recruitment groups receive. These can be caused by an excessive amount of teachers of some

groups that since they belong to the board the school principals have to assign them schedules. For

example, we can notice that the group 240, that recently had its need for teachers reduced (there used to

be two teachers in a classroom and now there is just one) has a substantially larger fraction of reductions

than the 350 Spanish group, that has only recently been getting more students.

For our planning to be fair to the different recruitment groups we should consider that all have equal

right to the reductions in teaching hours. For that we first estimate the Base Demand Teaching Hours,

which consists only in classes (for regular and non-regular education) and support activities:

Base Demand Teaching Hours = Teaching Hours Regular + Hours Support + Hours Support Teachers

+ Teaching Hours Non-Regular

(2.15)

24

Recruitment Group Fraction Effective Teaching Hours Fraction Reductions

200 - Portuguese and Social Studies/History 83.1% 17.6%210 - Portuguese and French 83.4% 16.9%220 - Portuguese and English 84.9% 14.6%230 - Mathematics and Nature’s Sciences 87.4% 12.9%240 - Visual and Technological Education 77.9% 20.7%250 - Musical Education 77.0% 18.0%260 - Physical Education 77.0% 22.0%290 - Moral and Catholic Education 87.3% 11.3%300 - Portuguese 83.6% 16.9%310 - Latin and Greek 83.1% 16.9%320 - French 83.2% 17.1%330 - English 82.4% 17.1%340 - German 59.8% 36.7%350 - Spanish 93.7% 6.5%400 - History 80.7% 19.7%410 - Philosophy 82.2% 18.2%420 - Geography 83.3% 17.1%430 - Economy and Accounting 83.6% 18.1%500 - Mathematics 88.3% 12.2%510 - Physics and Chemistry 88.2% 11.9%520 - Biology and Geology 86.4% 13.6%530 - Technological Education 73.8% 26.5%540 - Electrotechnical 92.6% 9.4%550 - Informatics 84.6% 14.9%560 - Agricultural and Livestock Sciences 73.8% 27.6%600 - Visual Arts 87.2% 12.8%610 - Musics 94.0% 6.2%620 - Physical Education 81.7% 18.0%910 - Special Education 1 96.7% 4.9%920 - Special Education 2 94.2% 7.7%930 - Special Education 3 97.0% 4.0%Average 84.9% 15.2%

Table 2.18: Fractions of the teaching hours that could be given that are actually given and that arereduced from the teacher’s teaching hours.

The teaching hours for regular education (Teaching Hours Regular ) will be obtained by multiplying

the number of classes for the teaching hours required per class (Teaching Hours Per Class) in tables

2.13, 2.14 and 2.15.

We assumed that the hours for support of students and support of other teachers in their classes

(Hours Support and Hours Support Teachers) were proportional to the number of classes. By dividing

the 2013/2014 Hours Support and Hours Support Teachers of a given recruitment group for the number

of classes in 2014 (predicted by the model) associated with that recruitment group we obtain the required

support hours per class (Required Support Hours Per Class).

For the Teaching Hours Non-Regular we assumed that their demand would be constant and equal

to its value in 2013/2014.

The Base Demand Teaching Hours is then:

Base Demand Teaching Hours = (Teaching Hours Per Class + Required Support Hours Per Class)NC

+ Teaching Hours Non-Regular

(2.16)

To calculate the total demand for teaching hours (Demand Teaching Hours) the base demand is

multiplied by the reductions component factor (Reduction Component Factor ). This is calculated by

dividing the Fraction Reductions by the Fraction Effective Lecture Hours and adding one. It is then

25

1 + 6.094.2 = 1.064 for the groups with single teacher and 1 +15.284.9 = 1.178 for the remaining ones.

Demand Teaching Hours = Base Demand Teaching Hours ∗ Reduction Component Factor (2.17)

After discussing this matter with the DGEEC direction we came to the conclusion that a 5 year transi-

tion period should be used after the hiring of teachers started in which the Reduction Component Factor

would linearly change from the present one for each teaching specialty to the average one presented

before.

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2.4 Teaching Hours from Board Teachers

Since we have already obtained the teaching hours needs we now need to estimate the teaching hours

offered by board teachers. The difference between the two will represent the unfulfilled needs of teaching

hours (or excessive ones).

2.4.1 Board Teacher Model

Our goal now is to develop a model to simulate the aging and retirement of board teachers. Since the

teacher data was only individualized from 2007/2008 to 2012/2013 then these will be the years used to

test and calibrate this model.

The simplest model that describes the system is a sequence of stocks that aggregate teachers by

age groups. We decided to create groups of 5 years because to plan how many teachers are needed

we don’t need information about yearly fluctuations, only about the general behavior for which 5 year

groups are adequate. The model schematics is displayed in figure 2.12.

Figure 2.12: Stock and flow diagram that models the aging and retirement of board teachers.

Considering that almost no new board teachers were hired between 2007/2008 and 2012/2013 we

decided not to add any teacher inflows.

It was considered that the teachers under 25 would become 25 in an average time of one year

(following an exponential distribution).

During this time interval the retirement age was increased linearly (6 months per year) from 60 years

in 2005 to 65 in 2015 (Jornal de Notı́cias, 2005). Some other changes were applied in 2013 and 2014

that further increased the retirement age from 65 to 66 (considered in section 2.4.2) and that encouraged

or discouraged early retirements (section 2.4.2). To simplify our first model we will start by considering

the retirement age to be constant and equal to 65. To simulate this we considered that once teachers

were 65 they would retire after an average of 0.1 years.

The results for the recruitment group of 1st cycle teachers (code 110) are displayed in figure 2.13

and for the recruitment group of 2nd cycle Mathematics and Science (code 230) these are displayed in

figure 2.14.

We can see that some differences occur yearly but they practically cancel after 5 years. This phe-

nomena is caused by the teacher aggregation in 5 year classes and was already expected. For the ages

between 55 and 65 the model is generating results that are systematically above the measured data.

But since the real retirement age was not 65 this was also expected.

27

Figure 2.13: Model results (blue) compared with the measured data (red) for the number of boardteachers of the recruitment group of 1st cycle teachers in the public education system.

Figure 2.14: Model results (blue) compared with the measured data (red) for the number of boardteachers of the teaching specialty 2nd cycle Mathematics and Science in the public education system.

To explain these differences we included the possibility of retirements earlier than 65 (that in this

time interval accounted for a mix of standard and early retirements). To model these retirements we

assumed that a fraction of the teachers that advance in age group will immediately retire. This fraction

28

was determined through the use of historical data. Since we did not model deaths then this retirement

fraction also incorporates the deaths of teachers in those age groups. The modified model’s schematics

is displayed in figure 2.15.

Figure 2.15: Stock and flow diagram that models the aging and retirement (early or not) of board teach-ers.

The obtained results for the recruitment group 110 are displayed in figure 2.16 and for group 230 in

figure 2.17.

Figure 2.16: Model results (blue) compared with the measured data (red) for the number of boardteachers of the recruitment group of 1st cycle teachers in the public education system. It was consideredthat 50% of teachers that became 55 years old retired and that 67% of teachers that became 60 yearsold also retired.

Since the deviations were small we found the model’s accuracy reasonable, considering the 5 year

aggregation of teachers. The teaching hours offered by board teachers are then obtained by:

Teaching Hours from Board Teachers =∑

Age Groups

Number of Teachers in Age Group∗

Number of Lecture Hours per Teacher in Age Group

(2.18)

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Figure 2.17: Model results (blue) compared with the measured data (red) for the number of boardteachers of the teaching specialty 2nd cycle Mathematics and Science in the public education system.It was considered that 15% of teachers that became 55 years old retired and that 60% of teachers thatbecame 60 years old also retired.

To improve future predictions a few more effects will be discussed.

2.4.2 Other Effects

In this section we consider effects that are certain to happen, either because they already happened in

2014 or because they are imposed by legislative measures.

Increase in Retirement Age

The retirement age in Portugal also increased from 2014 onward to 66 years (Público, 2013) and so in

2014 we increased the average time a teacher stays in the career after being 65 year old to 1 year. The

obtained results for groups 110 and 230 are displayed in figures 2.18 and 2.19.

We can observe, as expected, an increase in the number of teachers over 65 starting in 2014.

Hiring of New Board Teachers

The Portuguese government has also organized two extra teacher recruitment programs. Since these

already took place we know exactly how many teachers were recruited. For the recruitment group

of 1st cycle teachers 34 teachers were recruited in 2014 (for the school year 2013/2014) and 169 in

2015 (2014/2015). For the recruitment group of 2nd cycle Mathematics and Science 42 teachers were

recruited in 2014 and 251 in 2015.

30

Figure 2.18: Model results (blue) compared with the measured data (red) for the number of boardteachers of the recruitment group of 1st cycle teachers in the public education system. The retirementage rose to 66 in 2014.

Figure 2.19: Model results (blue) compared with the measured data (red) for the number of boardteachers of the teaching specialty 2nd cycle Mathematics and Science in the public education system.The retirement age rose to 66 in 2014.

We decided to approximate the age of the recruited teachers to 40 years

teacher demand model for basic education - ulisboa · 2013 the european commission demanded the...

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